On unaveraged convergence of positive operators in Lebesgue space

Abstract:

Let T be a power-bounded positive conservative operator on ${L_1}$ of a $\sigma$-finite measure space. Let e be a bounded positive function invariant under the operator adjoint to T. Theorem. (1) $\smallint |{T^n}f| \cdot \;e \to 0$ implies (2) $\smallint |{T^n}f| \to 0$. If T and all its powers are ergodic, and T satisfies an abstract Harris condition, then(l) holds by the Jamison-Orey theorem for all integrable f with $\smallint f \cdot e = 0$, and hence also (2) holds for such f. A new proof of the Jamison-Orey theorem is given, via the ’filling scheme’. For discrete measure spaces this is due to Donald Ornstein, Proc. Amer. Math. Soc. 22 (1969), 549-551. If T is power-bounded, conservative and ergodic, and $0 < {f_0} = T{f_0}$, then ${f_0}\; \cdot \;e \in {L_1}$ implies ${f_0} \in {L_1}$, hence (2) implies that ${T^n}f$ converges for each $f \in {L_1}$. Theorem. Let T be a positive conservative contraction on ${L_1}$; then the class of functions $\{ f - Tf,f \in L_1^ + \}$ is dense in the class of functions $\{ f - Tf,f \in {L_1}\}$.